This Document Contains Chapters 8 to 9 Brealey 5CE Solutions to Chapter 8 1. NPVA = –100 + 40 × annuity factor(11%, 4 periods) = $24.10 NPVB = –100 + 50 × annuity factor(11%, 3 periods) = $22.19 Both projects are worth pursuing. 2. Choose the project with the higher NPV, project A 3. If r = 16%, then NPVA = $11.93 and NPVB = $12.29. Therefore, you should now choose project B. 4. IRRA = Discount rate at which 40 × annuity factor(r, 4 periods) = 100 IRRA = 21.86% IRRB = 23.38% 5. No. Even though project B has the higher IRR, its NPV is lower than that of project A when the discount rate is lower (as in Problem 1) and higher when the discount rate is higher (as in Problem 3). This example shows that the project with the higher IRR is not necessarily better. The IRR of each project is fixed, but as the discount rate increases, project B becomes relatively more attractive compared to project A. This is because B’s cash flows come earlier, so their present values fall less rapidly when the discount rate increases. 6. The profitability indexes are as follows: Project A 24.10/100 = .2410 Project B22.19/100 = .2219 In this case, with equal initial investments, both the profitability index and NPV will give projects the same ranking. This is an unusual case, however, since it is rare for initial investments to be equal. 7. Project A has a payback period of 100/40 = 2.5 years. Project B has a payback period of 2 years. 8-1 This Document Contains Chapters 8 to 9 8. Project A Year Cash Flow ($) Discounted Cash Flow ($) @ 11 percent Cumulative Discounted Cash Flow ($) 0 -100 -100 -100 1 40 36.04 -63.96 2 40 32.48 -31.48 3 40 29.24 -2.24 4 40 26.36 +24.12 NPV= 24.12 Assuming uniform cash flows across time, the fractional year can now be determined. Since the discounted cash flows are negative until year 3 and become positive by Year 4, the project pays back sometime in the fourth year. Note that out of the total discounted cash flow of $26.36 in Year 4, the first $2.24 comes in by 2.24/26.36 = 0.084 year. Therefore, the discounted payback period for Project A is 3.084 years. Project B Year Cash Flow ($) Discounted Cash Flow ($) @ 11 percent Cumulative Discounted Cash Flow ($) 0 -100 -100 -100 1 50 45.05 -54.95 2 50 40.60 -14.35 3 50 36.55 +22.20 NPV= 22.20 The discounted payback for Project B is 2 years + 14.35/36.55 = 2.39 years. 9. No. Despite its higher payback, Project A still may be the preferred project, for example, when the discount rate is 11% (as in Problems 1 and 2). Just as in problem 5, you should note that the payback period for each project is fixed, but that NPV changes as the discount rate changes. The project with the shorter payback period need not have the higher NPV. 10. NPV = −3,000 + 800 × annuity factor(10%, 6 years) = $484.21 At this discount rate, you should accept the project. You can solve for IRR by setting the PV of cash flows equal to 3,000 on your calculator and solving for the interest rate: PV = −3000; n = 6; FV = 0; PMT = 800; 8-2 compute i. The IRR is 15.34%, which is the highest discount rate before project NPV turns negative. 11. Payback = 2500/600 = 4.167 years, which is less than the cutoff. So the firm would accept the project. 12. NPV = −10,000 + 3000 1.10 + 3000 1.102 + 5000 1.103 + 5000 1.104 = $2,378.25 Profitability index = NPV/Investment = .2378 13. Project at 2 percent discount rate Year Cash Flow ($) Discounted Cash Flow ($) @ 2 percent Cumulative Discounted Cash Flow ($) 0 -3000 -3000 -3000 1 800 784 -2216 2 800 768.8 -1447.2 3 800 753.6 -693.6 4 800 739.2 45.6 5 800 724.8 770.4 6 800 710.4 1480.8 NPV= 1480.8 Since the discounted cash flows become positive by Year 4, the project pays back sometime in the fourth year. Note that out of the total discounted cash flow of $739.20 in Year 4, the first $693.60 comes in by 693.60/739.20 = 0.94 year. Therefore, the discounted payback for the project is 3.94 years, and thus the project should be pursued. Project at 12 percent discount rate Year Cash Flow ($) Discounted Cash Flow ($) @ 12 percent Cumulative Discounted Cash Flow ($) 0 -3000 -3000 -3000 1 800 714.4 -2285.6 2 800 637.6 -1648.0 3 800 569.6 -1078.4 4 800 508.8 -569.6 5 800 453.6 -116.0 6 800 405.6 289.6 NPV= 289.6 8-3 Since the discounted cash flows become positive by Year 6, the project pays back in 5 years + 116/405.6 = 5.28 years. Therefore, given the firm’s decision criteria of a discounted payback of 5 years or less, the project should not be pursued. As illustrated by the two scenarios above, the firm’s decision will change as the discount rate changes. As the discount rate increases, the discounted payback period gets extended. 14. NPV = −2.2 + .3 × annuity factor(r, 15 years) − .9/(1 + r)15 When r = 6%, NPV = −2.2 + 2.538 = $0.338 billion When r = 16%, NPV = −2.2 + 1.576 = −$0.624 billion 15. The IRR of project A is 25.69%, and that of B is 20.69%. However, project B has the higher NPV and therefore is preferred. The incremental cash flows of B over A are –20,000 at time 0 and +12,000 at times 1 and 2. The NPV of the incremental cash flows is $827, which is positive and equal to the difference in project NPVs. 16. NPV = 5000 + 4000 1.12 – (1.12)2 11,000 = –$197.70 Because NPV is negative, you should reject the offer. You should reject the offer despite the fact that IRR exceeds the discount rate. This is a “borrowing type” project with positive cash flows followed by negative cash flows. A high IRR in these cases is not attractive: You don’t want to borrow at a high interest rate. 17. a. r = 0 implies NPV = 6,750 + 4,500 + (-18,000) = $-6,750 r = 50% implies NPV = 6750 + 4500 1.5 + 1.52 −18,000 = $1,750 r = 100% implies NPV = 6750 + 4500 2 + 22000 −18, = $4,500 b. IRR = 33.333%, the discount rate at which NPV = 0. 18. NPV = 10,000 + 1.122 − 7,500 + 1.123 − 8,500 = $-2,029.08 which is negative. So the project is not attractive. 8-4 The IRR of the project is -7.80 %. Since the IRR of the project is less than the required rate of return of 12%, the project should not be accepted using this rule. In this instance, both the NPV and IRR rules are giving consistent decisions. If the decisions using the two rules were inconsistent then, on balance, we would use the NPV rule. 19. NPV9% = –20,000 + 4,000 × annuity factor(9%, 8 periods) = $2139.28 NPV14% = –20,000 + 4,000 × annuity factor(14%, 8 periods) = –$1,444.54 The IRR is 11.81%. To confirm this on your calculator, set PV = (−)20,000; PMT = 4000; FV = 0; n = 8, and compute i. The project will be rejected for any discount rate above this rate. 20. a. The present value of the savings is 100/r If r = .08, PV = 1,250 and NPV = –1,000 + 1,250 = $250 If r = .10, PV = 1,000 and NPV = –1,000 + 1,000 = $0 b. IRR = .10 or 10%. At this discount rate, NPV = $0. c. Payback = 10 years. d. Discounted Payback Year Cash Flow ($) Discounted Cash Flow ($) @ 8 percent Cumulative Discounted Cash Flow ($) 0 -1000 -1000 -1000 1 100 92.6 -907.4 2 100 85.7 -821.7 3 100 79.4 -742.3 4 100 73.5 -668.8 5 100 68.1 -600.7 6 100 63.0 -537.7 7 100 58.3 -479.4 8 100 54.0 -425.4 9 100 50.0 -375.4 10 100 46.3 -329.1 11 100 42.9 -286.2 12 100 39.7 -246.5 13 100 36.8 -209.7 14 100 34.0 -175.7 15 100 31.5 -144.2 8-5 16 100 29.2 -115.0 17 100 27.0 -88.0 18 100 25.0 -63.0 19 100 23.2 -39.8 20 100 21.5 -18.3 21 100 19.9 1.6 Discounted payback when cost of capital is 8 percent = 20 years +18.3/19.9 = 20.95 years. The NPV=0 when the cost of capital =10%. The savings are supposed to last forever. Therefore, there is no finite discounted payback period when cost of capital is 10%. 21. a. NPV of the two projects at various discount rates is tabulated below. NPVA = –20,000 + 8,000 × annuity factor(r%, 3 years) = –20,000 + 8,000 [1r – r( 1 1+r)3 ] NPVB = –20,000 + 25,000 (1 +1 r)3 Discount Rate NPVA NPVB 0% 4000 5000 2% 3071 3558 4% 2201 2225 6% 1384 990 8% 617 –154 10% –105 –1217 12% –785 –2205 14% –1427 –3126 16% –2033 –3984 18% –2606 –4784 20% –3148 –5532 From the NPV profile, it can be seen that Project A is preferred over Project B if the discount rate is above 4%. At 4% and below, Project B has the higher NPV. b. IRRA = 9.70% [PV = (–)20; PMT = 8; FV = 0; n = 3; compute i] IRRB = 7.72% [PV = (–)20; PMT = 0; FV = 25; n = 3; compute i] 8-6 22. We know that the undiscounted project cash flows must sum to the initial investment because payback equals project life. Therefore, the discounted cash flows are less than the initial investment, so NPV must be negative. 23. NPV = 100 + –60 1.12 + (1.12)2 –60 = –1.40 Because NPV is negative, you should reject the offer. This is so despite the fact that IRR exceeds the discount rate. This is a “borrowing type” project with a positive cash flow followed by negative cash flows. A high IRR in these cases is not attractive: You don’t want to borrow at a high interest rate. 24. a. Project A Year Cash Flow ($) Discounted Cash Flow ($) @ 10 percent Cumulative Discounted Cash Flow ($) 0 -5000 -5000.00 -5000.00 1 1000 909.09 -4090.91 2 1000 826.45 -3264.46 3 3000 2253.94 -1010.52 4 0.00 0.00 -1010.52 NPV= -1010.52 The payback period for Project A is 3 years. Project A does not pay back on a discounted basis since cumulative discounted cash flows remain negative until the end of Year 4. Project B Year Cash Flow ($) Discounted Cash Flow ($) @ 10 percent Cumulative Discounted Cash Flow ($) 0 -1000 -1000.00 -1000 1 0 0 -1000 2 1000 826.45 -173.55 3 2000 1502.63 1329.08 4 3000 2049.04 3378.12 NPV= 3378.12 The payback period for Project B is 2 years. The discounted payback period for Project B is 2 years + 173.55/1502.63 = 2.12 years. Project C Year Cash Flow ($) Discounted Cash Flow ($) @ 10 percent Cumulative Discounted Cash Flow ($) 8-7 0 -5000 -5000.00 -5000.00 1 1000 909.09 -4090.91 2 1000 826.45 -3264.46 3 3000 2253.94 -1010.52 4 5000 3415.07 2404.55 NPV= 2404.55 The payback period for Project C is 3 years. The discounted payback period for Project C is 3 years + 1010.52/3415.07 = 3.3 years. b. Only B satisfies the 2-year payback criterion. c. You would accept Project B d. Projects B and C Project NPV A -1010.52 B 3378.12 C 2404.55 e. False. Payback gives no weight to cash flows after the cutoff date. 25. a. Year: 0 1 2 3 4 Sales 100 110 120 130 Costs 30 35 40 45 Depreciation 50 50 50 50 Net income 20 25 30 35 b. Cash flow = Net income + Depreciation Year: 1 2 3 4 CF: 70 75 80 85 NPV = –$2.30. NPV is negative even though the book rate of return is greater than the discount rate. 26. a. Cash flow each year = $5,000 – $2,000 = $3,000 NPV = –10,000 + 3,000 × annuity factor(8%, 5 years) = 1,978.13 NPV is positive so you should pursue the project. 8-8 b. The accounting change has no effect on project cash flows, and therefore no effect on NPV. 27. a. The present values of the project cash flows (net of the initial investments) are: NPVA = –2100 + 2000 1.20 + 1200 (1.20)2 = $400 NPVB = –2100 + 1440 1.20 + 1728 (1.20)2 = $300 The initial investment for each project is 2100. Profitability index (A) = 400/2100 = 0.1905 Profitability index (B) = 300/2100 = 0.1429 b. If you can choose only one project, choose A for its higher profitability index. If you can take both projects, you should: Both have positive profitability index. 28. a. The less–risky projects should have lower discount rates. b. First, find the profitability index of each project. PV of Profitability Project Cash flow Investment NPV Index A 3.79 3 0.79 0.26 B 4.97 4 0.97 0.24 C 6.62 5 1.62 0.32 D 3.87 3 0.87 0.29 E 4.11 3 1.11 0.37 Then select projects with the highest profitability index until the $8 million budget is exhausted. Choose, therefore, projects E and C. c. All the projects have positive NPV. All will be chosen if there is no rationing. 29. a. NPVA = –18 + 10 × annuity factor(10%, 3 periods) = $ 6.87 NPVB = –50 + 25 × annuity factor(10%, 3 periods) = $12.17 8-9 Thus Project B has the higher NPV if the discount rate is 10%. b. Project A has the higher profitability index. Invest Profitability Index Project PV ment NPV (= NPV/Investment) A 24.87 18 6.87 0.38 B 62.17 50 12.17 0.24 c. A firm with a limited amount of funds available should choose Project A since it has a higher profitability index of 0.38, i.e., a higher “bang for the buck.” For a firm with unlimited funds, the possibilities are: i. If the projects are independent projects, then the firm should choose both projects. ii. However, if the projects are mutually exclusive, then Project B should be selected. It has the higher NPV. 30. NPV Discount Rate Project A Project B 2% 43.43 41.31 12% 16.47 17.69 a. If r = 2%, choose A b. If r = 12%, choose B c. The larger cash flows of project A tend to come later, so their present values are more sensitive to increases in the discount rate. 31. a. Assuming an opportunity cost of capital of 6% (1) Payback Period Project Cash Flows, Dollars Payback Period, C0 C1 C2 C3 C4 C5 years I -250000 12000 18000 18000 30000 250000 5.69 II -25000 15000 8000 6000 6000 500 3.33 8-10 Project II provides the lowest payback period, and thus is the project of choice under this decision criterion. (2) Discounted Payback Period Project I Year Cash Flow ($) Discounted Cash Flow ($) @ 6 percent Cumulative Discounted Cash Flow ($) 0 -250000.00 -250000.00 -250000.00 1 12000.00 11320.75 -238679.25 2 18000.00 16019.94 -222659.31 3 18000.00 15113.15 -207546.16 4 30000.00 23762.81 -183783.35 5 250000.00 186814.54 3031.19 NPV= 3031.19 Discounted payback period is 4 years + 183783.35/186814.54 = 4.98 years. Project II Year Cash Flow ($) Discounted Cash Flow ($) @ 6 percent Cumulative Discounted Cash Flow ($) 0 -25000.00 -25000.00 -25000.00 1 15000.00 14150.94 -10849.06 2 8000.00 7119.97 -3729.09 3 6000.00 5037.72 1308.63 4 6000.00 4752.56 6061.19 5 500.00 373.63 6434.82 NPV= 6434.82 Discounted payback period is 2 years + 3729.09/5037.72 = 2.74 years. Project II has the lower discounted payback period at 2.74 years, and thus it is the best choice under this decision criterion. (3) NPV at 6% for: Project I = $3031.19 Project II = $6434.82 Since Project II offers a higher NPV, it is the best choice under this decision criterion. (4) IRRI = 6.29% IRRII = 19.04% Project II has the higher IRR and is, therefore, the better choice. 8-11 (5) Profitability Index (PI) = Initial Investmenti NPV PI Project I = 0.012 250,000 3,031.19 = PI Project II = 0.26 25,000 6,434.82 = Project II offers the highest ratio of net present value to investment; therefore, we would choose Project II. b. Of the two projects, Project II is the better choice. It has lower payback and discounted payback periods. In addition, Project II has the higher NPV, IRR and profitability index. 32. PV of Costs = 10,000 + 20,000 × annuity factor(12%, 5 years) = 10,000 + 72,096 = $82,096 The equivalent annual cost is the payment with the same present value, $22,774. [n = 5, i = 12, FV = 0; PV = (−)82,096; compute PMT] 33. Buy: PV of Costs = 80,000 + 10,000 × annuity factor(12%, 4 years) − 20,000/(1.12)4 = 80,000 + 30,373 − 12,710 = $97,663 The equivalent annual cost is the payment with the same present value, $32,154. [n = 4, i = 12, FV = 0; PV = (−)97,663; compute PMT] If you can lease instead for $30,000, then this is the less costly option. You also can compare the PV of the lease costs to the total PV of buying: 30,000× annuity factor(12%, 4 years) = $91,120 which is less than the PV of costs when buying the truck 34. a. The following table shows the NPV profile of the project. NPV is zero at an interest rate between 7% and 8% and at an interest rate between 33% and 34%. These are the two IRRs of the project. You can use your calculator to confirm that the two IRR’s are, more precisely, 7.16% and 33.67%. 8-12 Discount rate NPV Discount rate NPV 0.00 –2.00 0.21 0.82 0.01 –1.62 0.22 0.79 0.02 –1.28 0.23 0.75 0.03 –0.97 0.24 0.71 0.04 –0.69 0.25 0.66 0.05 –0.44 0.26 0.60 0.06 –0.22 0.27 0.54 0.07 –0.03 0.28 0.47 0.08 0.14 0.29 0.39 0.09 0.29 0.30 0.32 0.10 0.42 0.31 0.24 0.11 0.53 0.32 0.15 0.12 0.62 0.33 0.06 0.13 0.69 0.34 –0.03 0.14 0.75 0.35 –0.13 0.15 0.79 0.36 –0.22 0.16 0.83 0.37 –0.32 0.17 0.85 0.38 –0.42 0.18 0.85 0.39 –0.53 0.19 0.85 0.40 –0.63 0.20 0.84 0.41 –0.74 b. At 5% the NPV is: NPV = –22 + 20 1.05 + 20 1.052 + 20 1.053 – 40 1.054 = –0.443 Since the NPV is negative the project is not attractive. c. At 20% the NPV is: NPV = –22 + 20 1.20 + 20 1.202 + 20 1.203 – 40 1.204 = 0.840 At 40% the NPV is: NPV = –22 + 20 1.40 + 20 1.402 + 20 1.403 – 40 1.404 = –0.634 d. At a low discount rate, the positive cash flows ($20 for 3 years) are not discounted much. However, the final negative cash flow of $40 does not get discounted very heavily either. The net effect is a negative NPV. 8-13 At very high rates, the positive cash flows are discounted very heavily, resulting in a negative NPV. For mid-range discount rates, the positive cash flows that occur in the middle of the project dominate and project NPV is positive. 35. a. Econo-cool costs $300 and lasts for 5 years. The annual rental fee with the same PV is $102.53. We solve PMT × annuity factor(21%,5 years) = $300 PMT × 2.92598 = 300 PMT = $102.53 The equivalent annual cost of owning and running Econo-cool is $102.53 + $150 = $252.53 Luxury Air costs $500, and lasts for 8 years. Its equivalent annual rental fee is found from PMT × annuity factor(21%,8 years) = $500 PMT = $134.21 The equivalent annual cost of owning and operating Luxury Air is $134.21 + $100 = $234.21 b. Luxury Air is more cost effective. It has the lower equivalent annual cost. c. The real interest rate is now 1.21/1.10 – 1 = .10 = 10% Redo (a) and (b) using a 10% discount rate. Because energy costs would normally be expected to inflate along with all other costs, we should assume that the real cost of electric bills is either $100 or $150, depending on the model. Equiv. annual real cost to own Econo-cool = $ 79.14 plus $150 (real operating cost) = 150.00 $229.14 Equiv. annual real cost to own Luxury Air = $ 93.72 plus $100 (real operating cost) = 100.00 $193.72 Luxury Air is still more cost effective. 8-14 36. Time until NPV at purchase Cost purchase datea NPV todayb 0 400 – 31.33 –31.33 1 320 48.67 44.25 2 256 112.67 93.12 3 204.80 163.87 123.12 4 163.84 204.83 139.90 5 131.07 237.60 147.53 6 104.86 263.81 148.91 7 83.89 284.78 146.14 Notes: a. – Cost + 60 × annuity factor(10%, 10 years) b. NPV at purchase date/(1.10)n NPV is maximized when you wait 6 years to purchase the scanner. 37. The equivalent annual cost of the new machine is the 4-year annuity with present value equal to $20,000. This is $7005. This can be interpreted as the extra yearly charge that should be attributed to the purchase of the new machine spread over its life. It does not yet pay to replace the equipment since the incremental cash flow provided by the new machine, $10,000 – $5000 = $5000, is less than the equivalent annual cost of the machine. 38. a. The equivalent annual cost (EAC) of the new machine over its 10-year life is found by solving EAC × annuity factor(5%, 10 years) = $20,000 EAC × 7.7217 = $20,000 Therefore, EAC = $2590. Together with maintenance costs of $2000 per year, the equivalent cost of owning and operating is $4590. The old machine costs $5000 a year to operate, and is already paid for. (We assume it has no scrap value and therefore no opportunity cost.) The new machine is less costly. You should replace. b. If r = 10%, the equivalent annual cost of the new machine increases to $3255, so the equivalent cost of owning and operating it is now $5255, which is higher than that of the old machine. Do not replace. Your answer changes because the higher discount rate implies that the opportunity cost of the money tied up in the forklift also is higher. 8-15 39. At the time of writing this, the most recent period is the second quarter of 2010. For the second quarter of 2010 business investment in Machinery and Equipment (M&E) increased by 6.7% to $106.3 billion. From last year the same quarter, business investment in M&E rose by 11.8%. The capacity utilization rate for the second quarter of 2010 in the food industry was 83.8%, paper was 88%, plastics and rubber products was 73.5%, and machinery was 74.2%. The capacity utilization rate can be an indicator of the likelihood of future capital spending. If the capacity utilization rate is close to 100%, then it is highly likely that capital spending will increase in order to further increase capacity. The information was compiled from ”The Daily”, Economic indicators, Summary tables published by Statistics Canada. This information is available through the Statistics Canada website by using the following web link: http://www.statcan.gc.ca/dai-quo/economic_indicators-indicateurs_economiques- eng.htm 40. a. Present Value = Cash Flow at end of year Discount Rate – growth rate PV = 5,000 0.10 – 0.05 = $100,000 NPV = –$80,000 + $100,000 = $20,000 b. Recall that the IRR is the discount rate that makes NPV equal to zero: – Investment + PV of cash flows when discounted at IRR = 0 – 80,000 + IRR5,000–0.05 = 0 Solving the above equation, we find that: IRR = 5,000/80,000 + .05 = .1125 = 11.25% 41. For harvesting lumber, the value-maximizing rule is to cut the tree when its growth rate equals the discount rate. When the tree is young and the growth rate exceeds the discount rate, it pays to wait: the value of the tree is increasing faster than the discount rate. When the tree is older and the growth rate is less than r, cutting immediately is better, since the revenue from the tree can be invested to earn a rate of r, which is better than the tree is providing. 42. a. Time Cash flow 0 − 5 1 30 8-16 2 −28 The following graph shows a plot of NPV as a function of the discount rate. NPV = 0 when r equals (approximately) either 15.61% or 384%. These are the two IRRs. b. Discount rate NPV Develop? 10% −.868 million No 20% .556 Yes 350% .284 Yes 400% −.120 No 43. (a) PV of net cost = PV 12%,t – PV of salvage PV of Ultra Fast = 60,814.33 - [9,000/(1.12)5] = $55,707.49 PV of Medium Fast = 72,780.12 - [6,250/(1.12)4] = $68,808.13 EFFECTIVE ANNUAL COST Ultra Fast = $55,707.49/PV factor =$55,707.49 /3.605 = $15,452.84 Medium Fast= $68,808.13/PV factor =$68,808.13/3.037 = $22,656.61 Recommendation – buy Ultra Fast which costs less. b) Effective Annual Cost without salvage value Costs 0 1 2 3 4 5 PV 12%, t Ultra Fast 50,000 3,000 3,000 3,000 3,000 3,000 60,814.33 Medium Fast 50,000 7,500 7,500 7,500 7,500 72,780.12 -4 -2 0 2 4 0% 50% 100% 150% 200% 250% 300% 350% 400% 450% 500% Discount rate NPV 8-17 Ultra Fast = 60,814.33/3.605 = $16,869.44 Medium Fast = 72,780.12/3.037 = $ 23,964.48 c) With salvage value: Ultra Fast’s effective annual cost at the end of 4 years = $53,392.38/3.037 = $17,580.63. This is lower than Medium Fast (EAC = $22,656.94). Hence, we should purchase Ultra Fast which will be replaced by Hyper 3MM in 4 years. Without salvage value: Ultra fast EAC at the end of 4 years =$59,112.05/3.037= $19,463.96. This is still lower than medium fast (EAC = $23,964.48). 44. (a) Payback period: Year Project A cash flow A’s Cumulative cash flow Project B cash flow B’s cumulative cash flow 0 - 10,000 -10,000 -15,000 -15,000 1 6,000 -4,000 3,000 -12,000 2 6,000 2,000 5,000 -7,000 3 6,000 8,000 7,000 0 4 6,000 14,000 8,000 8,000 Payback for Project A = 1 + 4,000/6,000 = 1.667 years Payback for Project B = 3 years According to this method, Project A is preferred since it pays back earlier than Project B. While the payback method is relatively easy to use, its major disadvantages are: (i) It does not take into consideration the time value of money and (ii) it ignores cash flows beyond the payback period. Discounted Payback and NPV: Year Discount Factor 10 % A’s discounted cash flow A’s Cumulative Discount cash flow B’s discounted cash flow B’s cumulative Discount cash flow 0 1.000 -10,000 -10,000 -15,000 -15,000 1 .909 5,454 -4,546 2,727 -12,273 2 .826 4,956 410 4,130 -8,143 3 .751 4,506 4,916 5,257 -2,886 8-18 4 .683 4,098 9,014 5,464 2,578 NPV 9,014 2,578 Based on NPV project A is preferred. Discounted payback for Project A = 1 + (4,546/4,956) = 1.92 years Discounted payback for Project B = 3 + (2,886/5464) = 3.53 years Once again, according to the discounted payback method, Project A is preferred since it pays back earlier than Project B. The main advantage of the discounted payback period is that this method considers the time value of money, unlike the payback period. The main flaw of the discounted payback period is that it does not consider cash flows beyond the payback period and therefore, it may on occasion incorrectly reject positive NPV projects. (b). We see from the computations above that the NPV for Project A is $9,014 whereas for Project B it is $2,578. In general, notice also that if projects are able meet the cutoff in terms of a discounted payback, they must have a positive NPV. Based on the NPV approach, Project A is preferred to Project B as it has the higher NPV. (c). The profitability index for Project A = $9,014/$10,000 = .9014 For Project B, it is $2,578/$15,000 = .172. Once again, Project A is preferable using this method. (d). Internal Rate of return- Trial and Error Approach.: Essentially, with this approach we try to select the discount rate at which the IRR for the project =0. This discount rate is, then, also the project’s IRR. Project A Let us try a discount rate of 48 %. At this rate, NPV = -10,000 + {6,000/ (1+.48)1} + {6,000/ (1+.48)2} + {6,000/ (1+.48)3} + {6,000/ (1+.48)4} NPV = - 105.28 8-19 Since NPV is negative at this rate, the IRR should be lower than 48 percent. Let us try a discount rate of 46 %. At this rate, NPV = -10,000 + {6,000/ (1+.46)1} + {6,000/ (1+.46)2} + {6,000/ (1+.46)3} + {6,000/ (1+.46)4} NPV = 172.82 So, we can assume that the IRR for project A is somewhere between 46% and 48 %. Notice that this 2 % difference between the two rates has an “NPV distance” of 278.1 (i.e. 105.28 + 172.82). So, by interpolation, NPV will be 0 at a rate which is (2/278.1 x 105.28) below 48%; or 0.76% below 48% = 47.24% It is actually 47.23 % (using a financial calculator). Project B Let us try an initial discount rate of 17 %. At this rate, NPV = -15,000 + {3,000/ (1+.17)1} + {5,000/ (1+.17)2} + {7,000/ (1+.17)3} + {8,000/ (1+.17)4} NPV = - 143.54 Since NPV is negative at 17%, the IRR should be lower than this rate. Let us try a discount rate of 15 %. At this rate, NPV = - 15,000 + {3,000/ (1+.15)1} + {5,000/ (1+.15)2} + {7,000/ (1+.15)3} + {8,000/ (1+.15)4} NPV = 566.04 So, we know that the IRR for project B is some where between 15 % and 17 %. Once again, notice that this 2 % difference between the two rates has an “NPV distance” of 709.58 (i.e. 143.54 + 566.04). So, by interpolation, NPV will be 0 at a rate which is (2/709.58 x 143.54) below 17%; or 0.4% below 17% = 16.6% It is actually 16.58 % (using a financial calculator). Notice that, for both Projects A and B, we were able to get results for the IRR that were quite close to the actual numbers obtained through a financial calculator. 8-20 Using the IRR rule, Project A (with the higher IRR) is preferred over Project B. e) Independent projects would be evaluated for acceptance or rejection on a “standalone” basis. Mutually exclusive projects would be selected on an “either/or” basis and only those contributing most toward shareholder wealth would be selected. f) In this question we are able to reach the same decision using all the methods (payback, discounted payback, NPV, profitability index, and IRR) However, if we get conflicting decisions using different methods then the NPV method should be used as it is generally considered to be the most robust. 45. Assumptions Initial cash flow 100,000 First 2 years 0 Years 3 -7 0.15 Years 8 & 9 -0.02 Constant thereafter Discount rate 0.08 Years Cash flow Discounted cash flow Cumulative discounted cash flow 0 -100,000 1.000 -100000 (100,000.00) 1 0 0 (100,000.00) 2 0 0 (100,000.00) 3 16,000.00 0.7940 12,704.00 (87,296.00) 4 18,400.00 0.7350 13,524.00 (73,772.00) 5 21,160.00 0.6810 14,409.96 (59,362.04) 6 24,334.00 0.6300 15,330.42 (44,031.62) 7 27,984.10 0.5830 16,314.73 (27,716.89) 8 27,424.42 0.5400 14,809.19 (12,907.70) 9 26,875.93 0.5001 13,440.65 532.95 NPV after 9 years= $ 532.95 PV of cash flow into foreseeable future= 0.08 26,875.93 = $335,949.13 a) Project NPV = 532.95+335,949.13 = $336,482.08 Since the project NPV is positive, we should accept the project. b) The technique used in part (a) is the NPV decision rule. If a project has a positive NPV that is, the present value of future cash flow is greater than initial cost we accept that project. c) 8-21 Years Cash flow Cumulative cash flow Discounted cash flow Cumulative discounted cash flow -100,000.00 -100,000.00 -100000.00 (100,000.00) 1 0 -100,000.00 0 (100,000.00) 2 0 -100,000.00 0 (100,000.00) 3 16,000.00 (84,000.00) 12,704.00 (87,296.00) 4 18,400.00 (65,600.00) 13,524.00 (73,772.00) 5 21,160.00 (44,440.00) 14,409.96 (59,362.04) 6 24,334.00 (20,106.00) 15,330.42 (44,031.62) 7 27,984.10 7,878.10 16,314.73 (27,716.89) 8 27,424.42 35,302.52 14,809.19 (12,907.70) 9 26,875.93 62,178.45 13,440.65 532.95 1) Payback period = 6 + 27,984.10 20,106.00 = 6.72 yrs 2) Discounted payback = 8 + 13,440.65 12,907.70 = 8.96 yrs When we ignore the time value of money, we use the payback period. When we take into account the time value of money, we use the discounted payback period. The main advantage of using the payback period is that it is quick and easy to calculate. While the payback period is relatively easy to use, its major disadvantages are: (i).It does not take into consideration the time value of money and (ii) it ignores cash flows beyond the payback period. The discounted payback period is an improvement over the payback period to the extent that it considers the time value of money. However, to the extent that this method ignores cash flows beyond payback, it still suffers from a major flaw. d). Assume 150 years of constant cash flow after year 9 IRR= 17.13 % Assume 200 yrs of constant cash flow after year 9 IRR = 17.13 % 46. a) (i) Years Project Alpha cash flow Project Alpha cumulative cash flow Project Beta cash flow Project Beta cumulative cash flow 0 -100,000.00 -100,000.00 -100,000.00 -100,000.00 1 70,000.00 -30,000.00 40,000.00 -60,000.00 2 32,000.00 2000.00 40,000.00 -20,000.00 8-22 3 32,000.00 34,000.00 40,000.00 20,000.00 4 9,000.00 43,000.00 40,000.00 60,000.00 Project Alpha payback period = 1 + 32,000 30,000 = 1.94 years Project Beta payback period = 2 + 40,000 20,000 = 2.5 years Using the payback period, I would select Project Alpha which has the earlier payback period. ii) Years Discount factor Alpha discounted C.F Alpha Cum. cash flow Beta discounted C.F Beta Cum. cash flow 0 1.000 100,000.00 -100,000.00 100,000.00 -100,000.00 1 0.909 63,630.00 -36,370.00 36,360.00 -63,640.00 2 0.826 26,432.00 -9,938.00 33,040.00 -30,600.00 3 0.751 24,032.00 14,094.00 30,040.00 -560.00 4 0.683 6,147.00 20,241.00 27,320.00 26,760.00 Notice that we obtained discounted cash flows using the present value tables. If you do the calculations algebraically you are likely to get a somewhat different result. Alpha discounted payback = 2 + 24,032 9,938 = 2.41 yrs Beta discounted payback = 3 + 27,320 560 = 3.02 yrs Once again, using the discounted payback period, Project Alpha is preferred as it pays back earlier on a discounted basis than Project Beta. b) Alpha project NPV = $ 20,241.00 Beta project NPV = $ 26,760.00 Undertake Beta project using NPV decision rule since it has the higher NPV. c) IRR- Alpha = 22.4 % IRR- Beta =21.86 % Select project Alpha using IRR rule (Alpha has the higher IRR). d) Profitability index – Alpha = NPV/ Initial cash flow = 100,000 20,241 = 0.202 Beta = NPV/Initial cash flow = 0.268 100,000 26,760 = 8-23 Using the profitability index also, we would select Project Beta. e) -9,000 -4,500 0 4,500 9,000 13,500 18,000 22,500 27,000 31,500 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Discount rate NPV Alpha Beta Crossover point – Discount rate = 20.271 % NPV = 3,019.00 Notice that if the cost of capital rate is higher than 20.27%, the NPV rule would prefer project Alpha over project Beta. However, if the discount rate is lower than 20.27%, project Beta has the higher NPV and is preferred. However, according to the IRR rule, project Alpha (with the higher IRR) will be preferred to project Beta. 47. Annual Savings $12,000.00 Annual savings on W.C $4,000.00 Tax rate 0.35 Cost of New Machine $75,000 Number of years (Useful life) 5 Salvage value of New Machine $9,000 IRR project Alpha 22.4 % IRR project Beta 21.86 % Cross over point 20.271 % 8-24 Cost of old Machine $70,000 Salvage value of old machine in 5 years $6,500 Depreciable life 5 Salvage value of old machine, if sold today $30,000.00 Cost of Capital 0.12 Change in capital cost at time t0 = $75,000.00 - $30,000.00 = $45,000.00 Change in Salvage Value (SV) at time t5 = $9,000.00 - $6,500.00= $2,500.00 (assuming no tax implications on salvage value) After tax savings in operating cost = $12,000.00(1 - 0.35) = $7,800.00 Savings in working capital =$4,000.00 Total Present Value of Change in Depreciation =$ 10,645.25 (see calculation below) Year 1 2 3 4 5 New machine Depreciation 15,000 15,000 15,000 15,000 15,000 Old machine Depreciation 14,000 14,000 Change in Depreciation 1,000 1,000 15,000 15,000 15,000 Depreciation Tax Shield 350 350 5,250 5,250 5,250 Discount Factor @ 12 % 0.893 0.797 0.712 0.636 0.567 PV of change in depreciation 312.55 278.95 3,738 3,339 2,976.75 Total PV of depreciation change 10,645.25 NPV = [12,000(1-.035)]*(PVIFA 12%,5) + [2,500(1 – 0.35)]*(PVIF12%,5) + 4,000(PVIFA12%,5) + 10,645.25 – 45,000 = 7,800 (3.605) + 2,500(.567) + 4,000(3.605) + 10,645.25 – 45,000 = $ 9,601.75 Since the cost savings from installing the new machine and replacing the old machine results in a positive NPV, the new machine should replace the old machine. 8-25 Solution to Minicase for Chapter 8 None of the measures in the summary tables is appropriate for the analysis of this case, although the NPV calculations can be used as the starting point for an appropriate analysis. The payback period is not appropriate for the same reasons that it is always inappropriate for analysis of a capital budgeting problem: cash flows after the payback period are ignored; cash flows before the payback period are all assigned equal weight, regardless of timing; the cutoff period is arbitrary. The internal rate of return criterion can result in incorrect rankings among mutually exclusive investment projects when there are differences in the size of the projects under consideration and/or when there are differences in the timing of the cash flows. In choosing between the two different stamping machines, both of these differences exist. The net present value calculations indicate that the Skilboro machines have a greater NPV ($2.56 million) than do the Munster machines ($2.40 million). However, since the Munster machines also have a shorter life, it is not clear whether the difference in NPV is simply a matter of longevity. In order to adjust for this difference, we can compute the equivalent annual annuity for each: Munster machines: $2.40 0.15 (1.15) 1 0.15 C 1 7 = × − × million C × annuity factor(15%, 7 years) = $2.40 million C × 4.16042 = $2.40 million ⇒ C = EAC = $0.57686 million Skilboro machines: $2.56 0.15 (1.15) 1 0.15 C 1 10 = × − × million C × annuity factor(15%, 10 years) = $2.56 million C × 5.01877 = $2.56 million ⇒ C = EAC = $0.51009 million Therefore, the Munster machines are preferred. Another approach to making this comparison is to compute the equivalent annual annuity based on the cost of the two machines. The cost of the Munster machine is $8 million, so that the equivalent annual annuity is computed as follows: 8-26 Munster machines: $8 0.15 (1.15) 1 0.15 C 1 7 = × − × million C × annuity factor(15%, 7 years) = $8 million C × 4.16042 = $8 million ⇒ C = EAC = $1.92288 million For the Skilboro machine, we can treat the reduction in operator and material cost as a reduction in the present value of the cost of the machine: $2.50938 0.15 (1.15) 1 0.15 PV $500,000 1 10 = = × − × million $12.5 million – $2.50938 million = $9.99062 million $9.99062 0.15 (1.15) 1 0.15 C 1 10 = × − × million C × annuity factor(15%, 10 years) = $9.99062 million C × 5.01877 = $9.99062 million ⇒ C = EAC = $1.99065 million Here, the equivalent annual cost is less for the Munster machines. Note that the differences in the equivalent annual annuities for the two methods are equal. (Differences are due to rounding.) 8-27 Brealey 5CE Solutions to Chapter 9 A General Note: Many of the questions, for which solutions are provided below, require only that the NPV or IRR or some other evaluation criterion be calculated. These questions have not asked that you make a decision based on such criteria. In Chapter 7, we discussed the decision rules when we use these criteria. For instance, a positive NPV project should be accepted whereas a project with a negative NPV should be rejected. These decision rules should generally be kept in mind while working on the solutions below. 1. Net income = ($74 − 42 − 10) − .35 × ($74 − 42 − 10) = $22 − $7.7 = $14.3 million • Revenues − cash expenses − taxes paid = $74 − $42 − $7.7 = $24.3 million • Net Profit + Deprec = $14.3 + $10 = $24.3 million • (Revenues − cash expenses) × (1 − T) + T × Deprec = $32 × .65 + .35 × $10 = $24.3 million 2. a. ∆NWC = ∆Acct Receivable + ∆Inventory − ∆Acct Payable = −$1,500 + $1,000 − $2,000 = −$2,500 b. Cash flow = $36,000 − $24,000 + $2,500 = $14,500 3. Net income = ($7 − 4 − 1) − .40 × ($7 − 4 − 1) = $2 − $0.8 = $1.2 million • Revenues − cash expenses − taxes paid = $3 − $0.8 = $2.2 million • Net Profit + Deprec = $1.2 + $1.0 = $2.2 million • (Revenues − cash expenses) × (1 − T) + T × Deprec = $3 × .60 + .40 × $1 = $2.2 million 4. While depreciation is a non-cash expense, it still has an impact on net cash flow because of its impact on taxes. Every dollar of depreciation reduces taxable income by one dollar, and thus reduces taxes owed by $1 times the firm’s marginal tax rate. In Canada, such tax savings can be generated by capital cost allowance (CCA) which, for most assets, is computed using the written-down value method. CCA is computed for asset classes rather than for individual assets. Also, in the first year of 9-1 the asset’s life, the half-year rule becomes applicable. The various unique features of the declining balance CCA system make it quite different from straight-line depreciation. Compared with straight-line depreciation, declining balance CCA will move the tax benefits in time, and thus provide a different present value of the tax shield, thereby altering the value of the project. 5. Gross revenues from new chip = 12 million × $25 = $300 million Cost of new chip = 12 million × $8 = $96 million. Lost sales of old chip = 7 million × $20 = $140 million Saved costs of old chip = 7 million × $6 = $42 million. Increase in cash flow = (300 – 96) – (140 – 42) = $106 million 6. Revenue $160,000 Rental costs 35,000 Variable costs 45,000 Depreciation 10,000 Pretax profit 70,000 Taxes (35%) 24,500 Net income $45,500 7. a. Net Profit + Depreciation = $45,500 + $10,000 = $55,500 b. Revenue – rental costs – variable costs – taxes = $160,000 – $35,000 – $45,000 – $24,500 = $55,500 c. (Revenue – rental costs – variable costs) × (1–.35) + .35 × (Depreciation) = ($160,000 – $35,000 – $45,000) × .65 + .35 × $10,000 = $52,000 + $3,500 = $55,500 8. Change in working capital = ∆Accounts receivable – ∆Accounts payable = ($4500 – $1200) – ($200 – $600) = $3,700 Cash flow = $16,000 – $9,000 – $3,700 = $3,300 9-2 9. Incremental cash flows are: b. The cash that could have been realized by selling the art. d. The reduction in taxes paid. 10. Capital investment: $1,000,000 CCA calculation: Year UCC CCA (5%) End of Year UCC 1 $1,000,000 $25,000 $975,000 2 975,000 48,750 926,250 3 926,250 46,313 879,937 4 879,937 43,997 835,940 5 835,940 41,797 794,143 6 794,143 39,707 754,436 Operating cash flows of the project for the next six years (figures in thousands of dollars). Year: 0 1 2 3 4 5 6 Capital Investment -1,000 Revenues 120 120 120 120 120 120 Operating Expenses: Direct production costs 40 40 40 40 40 40 Fixed maintenance costs 15 15 15 15 15 15 Pre-tax Profits 65 65 65 65 65 65 Tax @35% 22.75 22.75 22.75 22.75 22.75 22.75 Operating Cash Flow (excluding CCA Tax Shield) 42.25 42.25 42.25 42.25 42.25 42.25 CCA Tax Shield (CCA x 35%) 8.75 17.063 16.209 15.399 14.629 13.898 Total Cash Flow -1,000 51.000 59.313 58.459 57.649 56.879 56.148 11. a. CCA calculation for the first 3 years: Year UCC CCA (30%) End of year UCC 1 $40,000 $6,000 $34,000 2 34,000 10,200 23,800 3 23,800 7,140 16,660 9-3 b. If the company has other assets in class 46 and the equipment is sold after 3 years, the adjusted cost of disposal is the sale price of $20,000. This amount is then deducted from the UCC of asset class 46. If overall UCC remains positive, we do not have to worry about CCA recapture. If, however, overall UCC becomes negative, we consider CCA recapture. The firm’s after-tax proceeds from the sale are $20,000 – PV of CCA tax shield lost - (0.35 x amount of CCA recapture, if applicable). c. If no other assets exist in Class 46 and the equipment is sold after 3 years, the adjusted cost of disposal is the sale price of $20,000. Subtracting this amount from the UCC of asset class 46 ($16,660 - $20,000 = -$3,340), we arrive at a negative balance, and thus recaptured depreciation. This amount is now added back to taxable income and the UCC of the asset class becomes zero. At the time of sale, the present value of tax shields lost as a result of the sale is calculated as: = r×+ 0.30× 0.35 × 1+1+0.r5r 16,660 0.30 , where r is the cost of capital. The firm’s after-tax proceeds from the sale are thus $20,000 – (0.35 x 3,340) – PV of tax shields lost = $18,831 – PV of CCA tax shields lost. 12. a. If the office space would have remained unused in the absence of the proposed project, then the incremental cash outflow from allocating the space to the project is effectively zero. The incremental cost of the space used should be based on the cash flow given up by allocating the space to this project rather than some other use. b. One reasonable approach would be to assess a cost to the space equal to the rental income that the firm could earn if it allowed another firm to use the space. This is the opportunity cost of the space. 13. Cash flow = Net income + depreciation – increase in NWC 1.2 = 1.2 + .5 – ∆NWC ∆NWC = $0.5 million 14. Cash flow = profit – increase in inventory = $10,000 – $1,000 = $9,000 9-4 15. NWC2009 = $32 + $25 – $12 = $45 million NWC2010 = $35 + $30 – $25 = $40 million Net working capital has decreased by $5 million. 16. Depreciation per year = $40/5 = $8 million Current Book value of old equipment = $40 – (3 × $8) = $16 million Sales price = $18 million After-tax cash flow = $18 – .35 × ($18 – $16) = $17.3 million 17. CCA calculation for the new capital investment (figures in thousands of dollars): Year UCC CCA (25%) End of year UCC 1 $10,000 $1,250 $8,750 2 8,750 2,188 6,562 3 6,562 1,641 4,922 4 4,922 1,231 3,691 5 3,691 923 2,768 Since the project ends after 5 years, and the equipment is sold, the adjusted cost of disposal is $4 million, which is deducted from the UCC asset class, that is 2.768 – 4 = -$1.232 million. This results in a negative balance and recaptured depreciation. The after-tax cash flow from the sale = $4 million – (.35 x $1.232) – PV of CCA tax shield lost. This equals $3.569 million – PV of tax shields lost. 18. a. The UCC increases by $6,000 to the extent of the purchase of the new washer but decreases by $2,000 to the extent of sale of the old washer. The net effect is an UCC increase of $4,000. CCA calculations are as follows: Year UCC CCA (30%) End of year UCC 1 $ 4,000 $ 600 $ 3,400 2 3,400 1,020 2,380 3 2,380 714 1,666 4 1,666 500 1,166 5 1,166 350 816 6 816 245 571 9-5 All dollar values should be interpreted as incremental results from making the purchase. First, we calculate operating cash flows excluding CCA tax shields. Year: 1 – 6 Earnings from Savings (before CCA) 1,500 Tax (40%) 600 Cash Flow from Operations (excluding CCA) $900 Now we consider the effect of the CCA tax shield on Bottoms Up’s cash flows. Year: 0 1 2 3 4 5 6 Capital Investment -6,000 After-tax Cash Flow from Operations (excl. CCA) 0 900 900 900 900 900 900 Cash Flow from Sale of Old Equipment 2000 0 0 0 0 0 0 Total Cash Flow (excl. CCA) -4,000 900 900 900 900 900 900 CCA Tax Shield (CCA x .4) 0 240 408 286 200 140 98 Total Project Cash Flow -4,000 1,140 1,308 1,186 1,100 1,040 998 b. The project NPV is calculated in two phases. First, we compute the total present value of cash flows excluding the CCA tax shield: PV = -4,000 + 900 x annuity factor(15%, 6 years) = -$594.4. Second, we calculate the present value of the CCA tax shield: PV of CCA tax shield = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 , where S = 0 = ( ) 0 1 0.15 1 0.5 0.15 0.15 0.3 4000 0.3 0.4 − + + × + × × = $997.10 NPV = Total PV excluding CCA tax shields + PV of CCA tax shield = -$594.40 + $997.10 = $402.7 c. Using straight-line depreciation, net cash flow at time 0 remains -$4,000, but the net cash flow at times 1 through 6 becomes $1,300, which is calculated as follows: 9-6 Earnings before depreciation $1,500 Depreciation (6000/6 years) 1,000 Taxable income 500 Taxes (0.40) 200 Net Income 300 + Depreciation 1,000 Operating Cash Flow $1,300 NPV = -4,000 + 1,300 x annuity factor (15%, 6 years) = $919.83 IRR = 23.21% 19. If the firm uses straight-line depreciation, the present value of the cost of buying, net of the annual depreciation tax shield (which equals .40 × 1000 = 400), is: 6000 – 400 × annuity factor(15%, 6 years) = 4486.21 The equivalent annual cost, EAC, is therefore determined by: EAC × 6-year annuity factor = 4486.21 EAC × 3.7845 = 4486.21 EAC = $1185.42 Note: this is the equivalent annual cost of the new washer, and does not include any of the washer’s benefits. 20. a. The year-wise CCA for the new grill, over its expected life, is as follows: Year UCC CCA (30%) End of year UCC 1 $20,000 $3,000 $17,000 2 17,000 5,100 11,900 3 11,900 3,570 8,330 Operating cash flow contribution, excluding tax shields, for year 1 through 3 = Saving in energy expenses x (1 - .35) = $10,000 x (1 - .35) = $6,500. Now, we must consider the effect of the CCA tax shield on the project’s yearly cash flows. Year: 1 2 3 Contribution from saving in energy expenses 6,500 6,500 6,500 CCA Tax Shield (CCA x .35) 1,050 1,785 1,250 Total Operating Cash Flow 7,550 8,285 7,750 9-7 b. Total Cash Flow (0-3) = Operating CF + CF associated with investments. At time 0, the CF from the investment is -$20,000. At the end of year 3, the grill is sold for $5,000. Therefore, total cash flows are: Time Cash Flows ($) 0 -20,000 1 7,550 2 8,285 3 12,750 [=7,750 + 5,000] c. First, we compute present value of cash flows excluding the CCA tax shield: PV = -20,000 + 6,500 x annuity factor(12%, 3 years) + 5,000 x discount factor (12%, 3 years) = -$829.3. We next calculate the present value of the CCA tax shield: PV of CCA tax shield: = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 = ( ) (1 0.12)3 1 0.3 0.12 5000 0.3 0.35 1 0.12 1 0.5 0.12 0.12 0.3 20000 0.3 0.35 + × + × × − + + × + × × = $3,842.41 NPV = Total PV excluding CCA tax shields + PV of CCA tax shield = -$829.3 + $3, 842.41 = -$3,013.11 21. a. Initial investment = $50,000 + $8,000 for working capital (20% of 40,000) = $58,000 b. CCA for the first 5 years of the plant and equipment’s life is as follows: Year UCC CCA (25%) End of year UCC 1 $50,000 $6,250 $43,750 2 43,750 10,938 32,812 3 32,812 8,203 24,609 4 24,609 6,152 18,457 5 18,457 4,614 13,843 9-8 (In thousands of dollars) Year: 0 1 2 3 4 5 Sales 40 30 20 10 0 Expenses 16 12 8 4 0 = Profit before tax 24 18 12 6 0 -tax @ 40% 9.6 7.2 4.8 2.4 0 = Operating Cash Flow (excl. CCA tax shield) 14.4 10.8 7.2 3.6 0 For calculating project cash flows for each year, we will need to calculate the tax savings generated from the CCA tax shield. We do this by multiplying each year’s CCA by the firm’s tax rate (40% in this case). (in thousands of dollars) Year: 0 1 2 3 4 5 Capital investment -50.00 Initial investment in working capital - 8.00 Decrease in working capital from previous year 2.0 2.0 2.0 2.0 Operating Cash Flow (excluding CCA tax shield) 14.4 10.8 7.2 3.6 Total Cash Flow (excluding CCA tax shield) - 58.00 16.4 12.8 9.2 5.6 CCA tax shield (CCA x 0.40) 2.5 4.4 3.3 2.5 Total - 58.00 18.9 17.2 12.5 8.1 c. The project NPV is calculated in two phases. First, we calculate the present value from cash flows excluding the CCA tax shield: Year: 0 1 2 3 4 Total Cash Flow (excluding CCA tax shield) (58) 16.40 12.80 9.20 5.60 x Discount Factor (10%) 1.000 0.909 0.826 0.751 0.683 PV of total cash flow (excl. CCA tax shield)* (58) 14.91 10.57 6.91 3.83 Total PV (excl. CCA tax shield) (21.78) 9-9 * Notice, you could also calculate this as follows, keeping in mind that there could be some difference of result due to rounding errors. (1.1.)82 (19..12)3 (15.1.6)4 12 1.1 − 58 + 16.4 + + + We next calculate the present value of the CCA tax shield: PV of CCA tax shield = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 , where S = 0 = ( ) 0 1 0.10 1 0.5 0.10 0.10 0.25 50000 0.25 0.4 − + + × + × × = $13,636 NPV (in thousands of dollars) = Total PV excluding CCA tax shields + PV of CCA tax shield = -$21.78 + $13.64 = -$8.14 22. a. The present value of costs from buying is $25,000 – $5000/(1.10)5 = $21,895 The cost of leasing (assuming that lease payments come at the end of each year) is $5,000 × annuity factor(10%, 5 years) = $18,954 Leasing is less expensive. b. The maximum lease payment, L, would be chosen so that L × annuity factor(10%, 5 years) = $21,895 L = $5,776 23. The initial investment is $100,000 for the copier + $10,000 in working capital, for a total outlay of $110,000. Depreciation expense each year = ($100,000 − $20,000)/5 = $16,000 9-10 The project saves $20,000 in annual labour costs, so its net operating cash flow including the depreciation tax shield is: $20,000 × (1 − .35) + .35 × $16,000 = $18,600 In year 5, the copier is sold for $30,000, which generates net-of-tax proceeds of $30,000 − .35 × $10,000 = $26,500 Note: tax is calculated on $10,000 which is the difference between the sale price of $30,000 and current book value of $20,000 In addition, the working capital associated with the project is freed up, which releases another $10,000 of cash. So non-operating cash flow in year 5 totals $36,500. The NPV is thus NPV = −110,000 + 18,600 × annuity factor(8%, 5 years) + 36,500/(1.08)5 = −110,000 + 99,106 = −$10,894 Because NPV is negative, Kinky’s should not buy the new copier. 24. Year: 0 1 2 3 4 5 Sales revenue 33,000 38,500 44,000 55,000 55,000 Less: cost 19,500 22,750 26,000 32,500 32,500 Profit before tax 13,500 15,750 18,000 22,500 22,500 Tax (35 percent) 4,725 5,513 6,300 7,875 7,875 Cash flow from operations (excluding CCA) (A) 8,775 10,237 11,700 14,625 14,625 Net working capital requirement 6,600 7,700 8,800 11,000 11,000 0 Investment in net working capital 6,600 1,100 1,100 2,200 0 -11,000 Investment in plant and equipment 25,000 Investment cash flow (B) 31,600 1,100 1,100 2,200 0 -11,000 Total cash flow (excluding CCA)(A – B) -31,600 7,675 9,137 9,500 14,625 25,625 Present value of total cash flow (excluding CCA) -31,600 (1.15) 7,675 (1.15)2 9,137 (1.15)3 9,500 (1.15)4 14,625 (1.15)5 25,625 Present value (excluding CCA) -31,600 6,674 6,909 6,246 8,362 12,740 9-11 = 9,331 9-12 Present value of CCA Tax Shield (PVTS), given a zero salvage value: + + × + × × = 1 0.15 1 (0.5 0.15) 0.15 0.15 25,000 0.15 0.35 = $4,090 NPV = $9,331 + $4,090 = $13,421 25. Find the equivalent annual cost of each alternative. Quick and Dirty Do-It-Right Operating Cost $ 1 million $ 1 million Investment $ 10 million $ 12 million Project Life 5 years 8 years PV CCA tax shield $ 2.37 million $ 2.84 million Net Capital Cost * $ 7.63 million $ 9.16 million EAC of Net Capital Cost ** $2.12 million $1.84 million Computation: PV of CCA tax shield for Quick and Dirty: + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.3 10 0.3 0.35 2.37 1.12 1.06 0.42 1.05 = = PV of CCA tax shield for Do-It Right: + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.3 12 0.3 0.35 2.84 1.12 1.06 0.42 1.26 = = 9-13 EAC for Quick and Dirty: $7.63m = Annuity (3.605) Annuity = $2.12 3.605 7.63 = m EAC for Do-It-Right: $9.16m = Annuity (4.968) Annuity = = 4.968 9.16 $1.84m Since the operating costs are the same, the project with the lower EAC is cheaper. This is Do-It-Right. * Investment – PV of CCA tax shield ** Annuity discounted at 12%; number of years = project life 26. All figures in thousands of dollars 0 1 2 3 4 Net working capital $220 $300 $140 $ 50 $ 0 Investment in NWC 220 80 –160 – 90 –50 Investment in Plant & eq 200 0 0 0 0 Cash flow from –$420 –$ 80 +$160 +$ 90 +$ 50 investment activity (A) All figures in thousands of dollars 0 1 2 3 4 Revenue $880 $1200 $560 $200 Cost 550 750 350 125 Pretax profit (excluding CCA) 330.00 450.00 210.00 75.00 – Taxes (35%) 115.50 157.50 73.50 26.25 Operating cash flow $214.50 $292.50 $136.50 $ 48.75 (excluding CCA tax shield) (B) Total CF –$420 $134.50 $452.50 $226.50 $98.75 (excluding CCA tax shield) (A+B) 9-14 Present Value of CCA Tax Shield (PVTS): + + × + × × = 1 .20 1 (0.5 0.20) 0.20 0.25 200 0.25 0.35 $35.65 1.2 1.1 0.45 17.5 = = NPV (in thousands of dollars): = PVTS + PV TOTAL CF (excluding CCA tax shield) = 2 3 (1.2)4 98.75 (1.2) 226.5 (1.2) 452.5 1.2 35.65 − 420 + 134.5 + + + = 35.65 – 420 + 112.08 + 314.24 + 131.08 + 47.62 = $ 220.67 27. All figures are on an incremental basis Labour savings $125,000 –Cost to run lathe 35,000 Net Savings (excluding CCA) 90,000 –Taxes (35%) 31,500 After tax savings (excluding CCA) $58,500 PV of CCA tax shield (PVTS): = + × + × × − + + × + × × (1 0.10)10 1 0.25 0.10 100,000 0.25 0.35 1 0.10 1 (0.5 0.10) 0.10 0.25 1,000,000 0.25 0.35 = − 0.35 × 0.3855 8,750 1.10 1.05 0.35 87,500 = 238,636.36 – 9,637.50 = $ 228,998.86 NPV = – 1,000,000 + 228,998.86 + 58,500 × annuity factor (10%, 10 years) + 100,000/(1.10)10 = -$372,987.71 9-15 28. You can access information on CCA asset classes and rates on commonly used assets by going to the following link on Revenue Canada’s website: http://www.cra-arc.gc.ca/tx/bsnss/tpcs/slprtnr/rprtng/cptl/clsss-eng.html As of October 30, 2010 this site has a table with 15 listed asset classes. The minimum eligible CCA rate is 4 percent and the maximum eligible rate is 100 percent. Thirteen of the 15 asset classes have declining balance CCA rates while asset Class 13 (leasehold interest) and asset Class 14 (patents, franchises, concessions or licenses for a limited period) involve straight-line computations. Notice that these classes include assets for which the cost to a business may not be a onetime initial outlay but rather a fixed recurring periodic cost over their economic life (such as, on leasehold interests). The CCA on such items is also computed as a fixed charge on a straight line basis. 29. Rogers Communication ($ million) 2009 2008 2007 Net capital expenditure. 1,015 2,025 1,023 Net Capital Expenditure to sales 8.7 % 17.9 % 10.1 % Sales & net capital expenditure to total assets 74.9 % 78.2 % 72.7 % Change in non-cash working capital items related to PP&E (55) 40 (20) Note: Calculations were done as follows: • Capital expenditure = change in gross Physical Plant &Equipment (PP&E) from year to year. For example, capital expenditure for 2007 = PP&E for 2007 minus PP&E for 2006. • Net capital expenditure (net Cap.Expd.) = capital expenditure – after tax sales of fixed assets. 30. If the savings are permanent, it is worth $250,000 to the firm. It can take $250,000 out of the project now without ever having to replace it. So the most the firm should be willing to pay is $250,000. 31. Project Evaluation Assumptions Plant and Equipment 100,000.00 Start up cost before tax 25,000.00 Start up cost after tax 16,500.00 # of years 5 Sales revenue year 1 60,000.00 Growth in sales: 1-4 5% Year 5 -5% Depreciation 20,000.00 Operating Exp 10,000.00 9-16 Tax rate 34% Cost of capital 12% 1 2 3 4 5 Sales 60,000.00 63,000.00 66,150.00 69,457.50 65,984.63 Operating cost - 10,000.00 - 10,500.00 - 11,025.00 - 11,576.25 - 10,997.44 Operating cash flow before tax 50,000.00 52,500.00 55,125.00 57,881.25 54,987.19 Taxes - 17,000.00 - 17,850.00 - 18,742.50 - 19,679.63 - 18,695.64 Operating cash flow (after tax) 33,000.00 34,650.00 36,382.50 38,201.63 36,291.54 Depreciation tax shield 6,800.00 6,800.00 6,800.00 6,800.00 6,800.00 Salvage value Total Cash Flow 39,800.00 41,450.00 43,182.50 45,001.63 43,091.55 (a) i) Note: Cash flow at year 0 includes initial investment after tax [100,000+ (25,000 *(1-.34)] Year Cash flow Cumulative cash flow 0 - 116,500.00 - 116,500.00 1 39,800.00 -76,700.00 2 41,450.00 -35,250.00 3 43,182.50 7,932.50 4 45,001.63 52,934.13 5 43,091.55 96,025.68 Payback Period = + 43,182.50 2 35,250.00 = 2.82 years Discount Payback Year Cash flow Discount Factor (12%) PV of cash flow 12 % Cumulative cash flow 0 - 116,500.00 1.000 - 116,500.00 - 116,500.00 1 39,800.00 0.893 35,541.40 - 80,958.60 2 41,450.00 0.797 33,035.65 - 47,922.95 3 43,182.50 0.712 30,745.94 -17,177.01 4 45,001.63 0.636 28,621.03 11,444.02 5 43,091.55 0.567 24,432.91 35,876.93 Discounted Payback period = + 28,621.03 3 17,177.01 = 3.6 years NPV = $35,876.93 IRR = 23.57 % Profitability Index = 0.31 116,500 35,876.93 = b). Using NPV and IRR decision rule the project should accepted. It has a positive NPV of $35,876.93 and an IRR of 23.57 % which is higher that the cost of capital rate. 9-17 (c) i) PV tax shield with zero salvage value = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 , where S = 0 + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.25 100,000 0.25 0.34 $21,742.28 1.12 1.06 0.37 8,500 = = NPV including CCA tax shield = $35,876.93 + $21,742.48 = $57,619.21 (ii) 1 2 3 4 5 Sales 60,000.00 63,000.00 66,150.00 69,457.50 65,984.63 Operating cost - 10,000.00 - 10,500.00 - 11,025.00 - 11,576.25 - 10,997.44 Operating cash flow before tax 50,000.00 52,500.00 55,125.00 57,881.25 54,987.19 Taxes - 17,000.00 - 17,850.00 - 18,742.50 - 19,679.63 - 18,695.64 Operating cash flow (after tax) 33,000.00 34,650.00 36,382.50 38,201.63 36,291.55 Depreciation tax shield 6,800.00 6,800.00 6,800.00 6,800.00 6,800.00 Salvage value 10,000.00 Total Cash Flow 39,800.00 41,450.00 43,182.50 45,001.63 53,091.55 Year Cash flow Discount Factor (12%) PV of cash flow 12 % Cumulative cash flow 0 - 116,500.00 1.000 - 116,500.00 - 116,500.00 1 39,800.00 0.893 35,541.40 - 80,958.60 2 41,450.00 0.797 33,035.65 -47,922.95 3 43,182.50 0.712 30,745.94 -17,177.01 4 45,001.63 0.636 28,621.03 11,444.02 5 53,091.55 0.567 30,102.91 41,546.93 NPV 41,546.93 PV of CCA tax shield with salvage value = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 = + × + × × − + + × + × × (1 0.12)5 1 0.25 0.12 10,000 0.25 0.34 1 0.12 1 (0.5 0.12) 0.12 0.25 100,000 0.25 0.34 = $20,438.73 NPV including CCA tax shield = $41,546.93 + $20,438.73 = $61,985.66 9-18 32. All cash flows are in millions of dollars. Sales price of machinery in year 5 is shown on an after-tax basis in year 5 as a positive cash flow on the capital investment line. Cash flow calculations are as follows: YEAR: 0 1 2 3 4 5 Sales (traps) 0.00 0.50 0.60 1.00 1.00 0.60 Revenue 0.00 2.00 2.40 4.00 4.00 2.40 Working capital 0.20 0.24 0.40 0.40 0.24 0.00 Change in Wk Cap 0.20 0.04 0.16 0.00 –0.16 –0.24 Revenue 0.00 2.0000 2.400 4.000 4.000 2.400 Expense 0.00 0.7500 0.900 1.500 1.500 0.900 Depreciation 0.00 1.2000 1.200 1.200 1.200 1.200 Pretax profit 0.00 0.0500 0.300 1.300 1.300 0.300 Tax 0.00 0.0175 0.105 0.455 0.455 0.105 After-tax profit 0.00 0.0325 0.195 0.845 0.845 0.195 CF from operations 0.00 1.2325 1.3950 2.0450 2.0450 1.3950 Cash flow CF: capital investments –6.00 0.0000 0.0000 0.0000 0.0000 0.3250 CF from wk cap –0.20 –0.0400 –0.1600 0.0000 0.1600 0.2400 CF from operations 0.00 1.2325 1.3950 2.0450 2.0450 1.3950 Total –6.20 1.1925 1.2350 2.0450 2.2050 1.9600 PV @ 12% –6.20 1.0647 0.9845 1.4556 1.4013 1.1122 Net present value –0.1817 33. If working capital requirements were only one-half of those in the previous problem, then the working capital cash flow forecasts would change as follows: Year 0 1 2 3 4 5 Original forecast –.20 –.04 –.16 0.0 .16 .24 Revised forecast –.10 –.02 –.08 0.0 .08 .12 Change in cash flow +.10 +.02 +.08 0.0 –.08 –.12 The PV of the change in the cash flow stream at a discount rate of 12% is $.0627 million. 34. a. Annual depreciation is (115 − 15)/5 = $20 million. Book value at the time of sale is $115 − (2 × $20) = $75 million Sales price = $80 million, so net-of-tax proceeds from the sale are: $80 − (.35 × $5) = $78.25 million 9-19 Therefore, the net cash outlay at time 0 is $150 − $78.25 = $71.75 million b. The project saves $10 million in expenses, and increases sales by $25 million. The new machine would entail depreciation of $50 million per year. Therefore, including the depreciation tax shield, operating cash flow increases by $35 × (1 − .35) + .35 × $50 = $40.25 million per year c. NPV = −71.75 + 40.25 × annuity factor(12%, 3 years) = $24.92 million. To find IRR, set the PV of the annuity to $71.75 and solve for the discount rate to find that IRR = 31.33%. d. All figures in $ millions After-tax annual operating cash flows: $35 × (1 – 0.35) = $ 22.75 million PV of after-tax operating cash flows: $35 × (1 – 0.35) × annuity factor (12%, 3) = $54.64 Net cash outlay at time 0: $150 – $80 = $70 million PV of net salvage value of new modem pool: $10.68 (1.12) ($30 $15) 1 − × 3 = PV of CCA tax shield: + × × − + + × + × × = 3 (1.12) 1 0.12 0.3 15 0.3 0.35 1 0.12 1 (0.5 0.12) 0.12 0.3 70 0.3 0.35 NPV 70 54.64 10.68 13.89 $9.21 million 16.56 2.67 $13.89 1.404928 1 0.42 1.575 1.12 1.06 0.42 7.35 = − + + + = = − = − = 35. Project evaluation 9-20 Note: 1. The 1-year feasibility study is a sunk cost and should not be considered. 2. Price/volume increase factor = (1+ inflation)*(1+ unit sales increase) = (1.015)*(1.04)= 1.0556 For example to find sale revenue in year 2, we multiply year 1 revenue by the price/volume factor. T1 T2 T3 T4 T5 T6 Sales Revenue 255,000 269,178 284,144 299,942 316,619 334,223 Less: Variable cost 16,000 16,889.6 17,828.7 18,819.9 19,866.3 20,970.9 Fixed cost 40,000 40,000 40,000 40,000 40,000 40,000 EBIT 199,000 212,288.4 226,315.3 241,122.1 256,758.7 273,252.1 Less: Taxes35 % 69,650 74,300.9 79,210.4 84,392.7 89,865.5 95,638.2 Net Income 129,350 137,987.5 147,104.9 156,729.4 166,893.2 177,613.9 T0 T1 T2 T3 T4 T5 T6 Net Working Capital 40,000 44,000 48,400 53,240 58,564 64,420 70,862 Change in NWC 4,000 4,400 4,840 5,324 5,856 6,442 T0 T1 T2 T3 T4 T5 T6 Investment: Land 150,000 Building 350,000 Equipment 250,000 Net working Capital 40,000 (790,000) ∆ NWC (4,000) (4,400) (4,840) (5,324) (5,856) (6,442) Net Income (Excluding CCA tax shield) 129,350 137,987.5 147,104.9 156,729.4 166,893.2 177,613.9 Salvage Value: Building 300,000 Equipment 125,000 T0 T1 T2 T3 T4 T5 T6 Total Cash flow (excluding CCA tax shield) (790,000) 125,350 133,587.5 142,264.9 151,405.4 161,037.2 596,171.9 Discount Factor (12%) 1.000 .8929 .7972 .7118 .6355 .5674 .5066 PV excluding CCA tax shield (790,000) 111,925 106,495.9 101,264.1 96,218.13 91,372.5 302,020.7 Total PV (excluding CCA tax shields) 19,296.33 9-21 PV of CCA tax shield: Building = + × + × × − + + × + × × (1 0.12)6 1 0.12 .04 300,000 0.04 0.35 1 0.12 1 (0.5 0.12) 0.12 0.04 350,000 0.04 0.35 = − 0.16 × 0.50663 4,200 1.12 1.06 0.16 4,900 = $15,685.34 Manufacturing Equipment = + × + × × − + + × + × × (1 0.12)6 1 0.25 0.12 125,000 0.25 0.35 1 0.12 1 (0.5 0.12) 0.12 0.25 250,000 0.25 0.35 = − 0.37.5 × 0.50663 10,937 1.12 1.06 0.37 21,875 = (59,121.62 x .94643) – 14,976.39 = $ 40,978.08 NPV = 19,296.33 + 15,685.34 + 40,978.08 = $ 75,959.75 Since the project has a positive net present value we should go ahead with it. 36. Assumptions Plant and Equipment 160,000.00 Building 40,000.00 Number useful life (yrs) 8 Sales revenue year 1 60,000.00 Growth in sales: 1-3 0 Growth in sales: 4-6 10 % Growth in sales: 6-8 - 5 % Depreciation P&E 20,000.00 Building 5,000.00 Operating Exp 15,000.00 Tax rate 34% Cost of capital 12% 1 2 3 4 5 6 7 8 Sales 60,000 60,000 60,000 66,000 72,600 79,860 75,867 72,074 Operating cost -15,000 - 15,000 -15,000 -16,500 -18,150 -19,965 -18,967 -18,018 Operating cash flow before tax 45,000 45,000 45,000 49,500 54,450 59,895 56,900 54,055 Taxes -15,300 - 15,300 -15,300 -16,830 -18,513 -20,364 -19,346 -18,379 Operating cash flow 29,700 29,700 29,700 32,670 35,937 39,531 37,554 35,676 9-22 (after tax) Dep. tax shield: P&E 6,800 6,800 6,800 6,800 6,800 6,800 6,800 6,800 Building 1,700 1,700 1,700 1,700 1,700 1,700 1,700 1,700 Total Cash Flow 38,200 38,200 38,200 41,170 44,437 48,031 46,054 44,176 (a) Years Cash flow Discount Factor (12%) PV of CF (12 %) Cumulative cash flow 0 - 200,000.00 1.00000 - 200,000.00 - 200,000.00 1 38,200.00 0.89286 34,107.14 - 165,892.86 2 38,200.00 0.79719 30,452.81 - 135,440.05 3 38,200.00 0.71178 27,190.01 - 108,250.04 4 41,170.00 0.63552 26,164.28 - 82,085.76 5 44,437.00 0.56743 25,214.75 - 56,871.01 6 48,030.70 0.50663 24,333.85 - 32,537.16 7 46,054.17 0.45235 20,832.57 - 11,704.59 8 44,176.46 0.40388 17,842.13 6,137.54 NPV 6,137.54 Based on the positive NPV Virtual Printing should accept the finance manager’s recommendations. (b) The technique used in part (a) is the net present value decision rule. Accordingly, accept projects with positive NPV because the total present value of future cash flows is greater than the initial cost. (c) i) Years Cash flow Cumulative cash flow 0 - 200,000.00 - 200,000.00 1 38,200.00 - 161,800.00 2 38,200.00 - 123,600.00 3 38,200.00 - 85,400.00 4 41,170.00 - 44,230.00 5 44,437.00 207.00 6 48,030.70 48,237.70 7 46,054.17 94,291.87 8 44,176.46 138,468.33 Payback period = 4 + (44,203/44437) = 4.995 years (ii) Discounted payback = (( )) 7.66years 17,842.13 7 + 11,704.59 = Advantage for payback period – It is relatively easy to use. Disadvantage for payback period – Does not take into consideration the time value of money. This method also ignores cash flows beyond the payback period. Advantage of discounted payback- this method considers time value of money, unlike payback period. Also, if the projects meet the cutoff, it must have a positive NPV. 9-23 Disadvantage of discounted payback – It does not consider cash flows beyond the payback period and therefore, it may incorrectly reject positive NPV projects. Also, it is not easier to use than NPV rule because both projected cash flow and discount rate must be determined. (d) IRR = 12.84 (e) Years Cash flow Discount Factor (12%) PV of cash flow (12% ) Cumulative cash flow 0 - 200,000.00 1.000 - 200,000.00 - 200,000.00 1 38,200.00 0.89286 34,107.14 - 165,892.86 2 38,200.00 0.79719 30,452.81 - 135,440.05 3 38,200.00 0.71178 27,190.01 - 108,250.04 4 41,170.00 0.63552 26,164.28 - 82,085.76 5 44,437.00 0.56743 25,214.75 - 56,871.01 6 48,030.70 0.50663 24,333.85 - 32,537.16 7 46,054.17 0.45235 20,832.57 - 11,704.59 8 54,176.46 0.40388 21,880.96 10,176.37 10,176.37 PV tax shield Building = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 , where S = 0 + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.04 10,000 0.04 0.34 = $804.46 PV tax shield P&E with salvage value = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 = + × + × × − + + × + × × (1 0.12)8 1 0.25 0.12 10,000 0.25 0.34 1 0.12 1 (0.5 0.12) 0.12 0.25 160,000 0.25 0.34 = 34,787.64 – 927.72 = $33,859.92 PV tax shield building = $804.46 Total NPV with salvage value = 33,859.92 + 804.46+ 10,176.37 = $ 44,840.75 9-24 Solution to Minicase for Chapter 9 The spreadsheet on the next page shows the cash flows associated with the project. Lines 1 – 11 match the data given in Table 9.11 except for the substitution of CCA. Line 8, capital investment, shows the initial investment of $1.5 million in refurbishing the plant and buying the new machinery. When the project is shut down after 5 years, the machinery and plant will be worthless. But they will not be fully depreciated and will continue to generate CCA tax shields assuming that Sheetbend has other assets in the respective asset classes. The present value of the CCA tax shields on the refurbished plant and new machinery are entered in lines 14 and 15, respectively. The working capital requirement is 10 percent of sales, or $300,000. This means, the investment in working capital (line 9) initially is $300,000, but in Year 5, when the project is shut down, the investment in working capital is recouped. If the project goes ahead, the land cannot be sold until the end of year 5. If the land is sold for $600,000 (as Mr. Tar assumes it can be), the taxable gain on the sale is .5 x $590,000 = $295,000, since the land is carried on the books at $10,000. Therefore, the cash flow from the sale of the land, net of tax at 35%, is $496750. The net present value of the project, which accounts for the present value of the total cash flows (Line 13) and the present value for CCA tax shield of the refurbished plant (Line 14) and the new machinery (Line 15), at a 12% discount rate, is $683,480 (Line 16). If the land can be sold for $1.5 million immediately, the after-tax proceeds will be 1,500,000 – .35 x .5(1,500,000 – 10,000) = $1,239,250 So it appears that immediate sale is the better option. However, Mr. Tar may want to reconsider the estimate of the selling price of the land in 5 years. If it can be sold today for $1,500,000 and the inflation rate is 4%, then perhaps it makes more sense to assume it can be sold in 5 years for 1,500,000 × 1.045 = $1,824,979. In that case, the forecasted after-tax proceeds of the sale of the land in 5 years rises to $1,507,357, which is $1,010,607 higher than the original estimate of $496,750; the present value of the proceeds from the sale of the land increases by $1,010,607/1.125 = $573,445. Therefore, under this assumption, the present value of the project increases from the original estimate of $683,480 to a new value of $1,256,925 and in this case the project is more valuable than the proceeds from selling the land immediately. The extent to which the project is now more valuable is $1,256,925 – $1,239,250 = $17,675. 9-25 Dollar figures in thousands, except price per yard. Year 0 1 2 3 4 5 1. Yards sold 100.00 100.00 100.00 100.00 100.00 2. Price per yard 30.00 30.00 30.00 30.00 30.00 3. Revenue 3000.00 3000.00 3000.00 3000.00 3000.00 4. Cost of goods sold 2100.00 2184.00 2271.36 2362.21 2456.70 5. Operating cash flow (excluding CCA) 900.00 816.00 728.64 637.79 543.30 6. Tax at 35% 315.00 285.60 255.02 223.23 190.16 7. Cash flow from operations after-tax (excluding CCA) 585.00 530.40 473.62 414.56 353.14 8. Capital investment –1500.00 9. Investment in working capital –300.00 300.00 10. Sale of land (after tax) 496.75 11. Total cash flow –1800.00 585.00 530.40 473.62 414.56 1149.89 12. PV of total cash flow (excluding CCA) –1800.00 522.32 422.83 337.11 263.45 652.45 13. Total present value of the cash flows (excluding CCA) (A) 398.16 14. PV of CCA tax shield on refurbished plant1 (B) 48.71 15. PV of CCA tax shield of new machinery1 (C) 236.61 16. Net present value (A) + (B) + (C) 683.48 1. The calculations for the CCA tax shields are as follows: PV of the CCA tax shield on the refurbished plant: = c c ( r)t d r SdT r r r d CdT + × − + + + + 1 1 1 1 0.5 , where S = 0 + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.05 500000 0.05 0.35 $48,713.24 1.12 1.06 0.17 8750 = = PV of the CCA tax shield of the new machinery: + + × + × × = 1 0.12 1 (0.5 0.12) 0.12 0.30 1000000 0.30 0.35 $236,607.14 1.12 1.06 0.42 105000 = = 9-26 (We compare the NPV of the project to the value of an immediate sale of the land. This treats the problem as two competing mutually exclusive investments: sell the land now versus pursue the project. The investment with higher NPV is selected. Alternatively, we could treat the after-tax cash flow that can be realized from the sale of the land as an opportunity cost at time 0 if the project is pursued. In that case, the NPV of the project would be reduced by the initial cash flow given up by not selling the land. Under this approach, the decision rule is to pursue the project if NPV is positive, accounting for that opportunity cost. This approach would result in the same decision as the one we have presented.) 9-27 Solution Manual for Fundamentals of Corporate Finance Richard A. Brealey, Stewart C. Myers, Alan J. Marcus, Elizabeth Maynes, Devashis Mitra 9780071320573, 9781259272011
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